Conserved and non-conserved Noether currents from the quantum effective action
aa r X i v : . [ h e p - t h ] F e b Prepared for submission to JHEP
Conserved and non-conserved Noether currents from thequantum effective action
Stefan Floerchinger and Eduardo Grossi Institut f¨ur Theoretische Physik Universit¨at Heidelberg,Philosophenweg 16, D-69120 Heidelberg Center for Nuclear Theory, Department of Physics and Astronomy, Stony Brook University, Stony Brook, NewYork 11794-3800, USA
E-mail: [email protected] , [email protected] Abstract:
The quantum effective action yields equations of motion and correlation functions including allquantum corrections. We discuss here how it encodes also Noether currents at the full quantum level. Thisholds both for covariantly conserved currents associated to real symmetries that leave the action invariantas well as for non-conserved Noether currents associated to extended symmetry transformations whichchange the action, but in a specific way. We discuss then in particular symmetries and extended symmetriesassociated to space-time geometry for relativistic quantum field theories. These encompass local dilatationsor Weyl gauge transformation, local Lorentz transformations and local shear transformations. Togetherthey constitute the symmetry group of the frame bundle GL( d ). The corresponding non-conserved Noethercurrents are the dilatation or Weyl current, the spin current and the shear current for which divergence-typeequations of motion are obtained from the quantum effective action. ontents The relation between the microscopic formulation of a quantum field theory, and the macroscopic formu-lation which includes the effect of quantum and statistical fluctuations, can be nicely discussed in terms ofactions. The microscopic action S [ χ ] defines a theory at a microscopic scale or at very high momenta wherequantum fluctuations are suppressed. This is the object that enters the functional integral. In classicalsituations where quantum fluctuations are negligible, the microscopic action yields directly the classicalEuler-Lagrange equations and for this reason it is sometimes called classical action. The microscopic actiondepends of course on the field values but is otherwise independent of the state. Initial conditions enteronly as boundary conditions for solutions to the equations of motion.In contrast to this, the one-particle irreducible or quantum effective action (see e. g. [1, 2]) dependson field expectation values and yields different expressions for which all quantum corrections have already– 1 –een taken into account. For example, the propagators and vertices obtained from functional derivativesof the one-particle irreducible effective action around a vacuum solution yield the full correlation functionsand S-matrix elements when used in tree diagrams [1, 2].Similarly, from the variation of the quantum effective action with respect to the fields one obtainsrenormalized equations of motion. It becomes already clear from these statements that the determinationof the quantum effective action itself is typically a formidable task. In particular, it differs from the micro-scopic action through both perturbative and non-perturbative quantum and statistical corrections. Thereare several methods to take these corrections into account; one of them is the functional renormalizationgroup [3–6].One should remark here that the quantum effective action can, and will, contain terms of higherderivatives in the fields and possibly non-local terms. All terms allowed by symmetries can appear andare typically present. Scaling arguments based on the classification of operators into relevant, irrelevantand marginal can sometimes be used in the vicinity of renormalization group fixed points, but that is notthe generic situation. One should also remark that the quantum effective action depends in general on thequantum state through the boundary conditions of the functional integral.In the present paper we are interested in the relation between Noether currents [7] and the quantumeffective action. Because it differs in a rather non-trivial way from the microscopic action, also the cor-responding Noether currents can differ substantially. In the presence of quantum anomalies it is possiblethat a current is conserved at the classical level, but not conserved when quantum fluctuations have beentaken into account.This motivates a detailed study of how Noether currents follow in full glory from the quantum effectiveaction. Both expectation values of currents, as well as their correlation functions, are actually of interest.One reason is fluid dynamics. Experience shows that in the macroscopic regime, i. e. for large time intervalsand long distances, one can approximate quantum field dynamics often rather well by a variant of fluiddynamics [8–10]. This follows the rational that those degrees of freedom are important over long timeintervals that are preserved from relaxation by conservation laws [11–13]. In practise, for a relativisticfluid, it is by reasons of causality not possible to take only the strict hydrodynamical degrees of freedom(which are directly governed by conservation laws, e. g. energy- and momentum density) into account,but some non-hydrodynamical fields must be propagated, as well [14–18]. For example, in Israel-Stewarttheory [19, 20] for a fluid with conserved energy-momentum tensor but no additional conserved quantumnumbers these are the shear stress and bulk viscous pressure. Equations of motion for the latter aretypically postulated in a phenomenological way. We will actually argue that these identities should followfrom the additional non-conservation laws for the dilatation current and shear current. In some regards,the equations we find are close to those proposed in ref. [21].An extension of fluid dynamic equations to include the spin tensor has been proposed and discussedin several recent publications [22–28], partly with the motivation to explain the measurement of the polar-ization of the hyperon Λ produced in high energy nuclear collisions [29] (see ref. [30] for a review).Correlation functions of various conserved and non-conserved currents are also of high interest. Forexample, they can be used to describe critical behaviour in the vicinity of phase transitions. In the context oflinear or weakly non-linear response theory around equilibrium states they can be used to obtain transportproperties, e. g. through the Kubo relations [31–39].In the present paper we have two main goals. The first is to establish a general formalism for obtainingcurrents associated to different symmetry transformations for the quantum effective action. Our focus will– 2 –e on expectation values of such currents, but the formalism can in a similar way also be used for correlationfunctions. Our discussion will include traditional symmetry transformations that leave the action invariant,but also what we call extended symmetry transformations [40–42]. The latter do not leave the actioninvariant but change it in a specific way by a term that is actually known in the macroscopic theory.We will argue that such transformations are still very powerful. Associated to such extended symmetrytransformations are non-conserved Noether currents.We will discuss different examples for normal and extended symmetries related to the geometry ofspace-time for a relativistic quantum field theory. These transformations encompass local changes ofcoordinates (diffeomorphisms), but also transformations in the frame and spin bundles such as local Lorentztransformations, dilatations, and shear transformations. The associated currents we discuss encompass twoversions of the energy-momentum tensor (a generalization of the canonical energy-momentum tensor to thequantum effective action and the symmetric energy-momentum tensor), the spin current, the dilatation orWeyl current, and the shear current. The latter three form together a tensor of rank three known as thehypermomentum tensor [43–47].Hypermomentum was initially investigated mainly in the context of modified theories of gravity, be-cause it constitutes natural sources for non-Riemannian geometrical structures such as torsion and non-metricity [43–57]. We will also essentially obtain the currents of hypermomentum from varying thesegeometrical structures and observing the response of the quantum effective action. However, we want toargue that hypermomentum is not only of interest in the context of somewhat speculative extensions ofEinsteins theory of general relativity. In fact we believe that the spin current, dilatation current and shearcurrent can all be non-zero for generic interacting quantum field theories in non-equilibrium situations.Even if these currents are apparently absent at the classical level, they can arise from quantum fluctuations,similar to quantum anomalies. We will also discuss an (approximate) effective action for a scalar field withnon-minimal coupling to gravity to gravity to underline this point. Symmetries play an important role in quantum field theory, for at least three reasons. First, they constrainsubstantially the form of a microscopic action S [ χ ] and define in this sense to a large extend a microscopictheory. Together with renormalizability, symmetries provide the main guiding principle for the constructionof microscopic physics theories.The second reason is that symmetries constrain also to a large extend the quantum effective actionΓ[ φ ] where φ = h χ i . One may think that this is less important because Γ[ φ ] is anyway derived from S [ χ ](we recall the construction below), but in practice the effective action Γ[ φ ] can usually not be obtainedexactly, so that insights gained through symmetry considerations are particularly important and powerful.In the absence of anomalies or explicit symmetry breaking, only terms that are allowed by the symme-tries are allowed to appear in the effective action. It is this aspect of symmetries (and extended symmetries)that will be discussed in the present section. The third important aspect of symmetries are the conservationlaws they lead to. This will be discussed in the subsequent section.While traditionally a symmetry is a transformation that leaves the action invariant, we will in thepresent paper also discuss more general transformations (or formally Lie group / Lie algebra actions)for which this is not the case, but that are nevertheless rather useful because they change the action in a– 3 –pecific way that allows to make powerful statements, as well. We refer to such transformations as extendedsymmetries (see below). We consider a theory for quantum fields χ ( x ) which we do not specify in further detail here. In practise χ ( x ) can stand for a collection of different fields and may encompass components transforming as scalars,vectors, tensors, or spinors. The theory is described by a microscopic action S [ χ ]. The latter enters the partition function Z [ J ] = Z Dχ e iS [ χ ] − i R x { J ( x ) χ ( x ) } . (2.1)In eq. (2.1) we have introduced sources J ( x ) for the fundamental fields χ ( x ). More generally one mayalso introduce sources for composite operators as will be discussed in more detail below. For example, themetric g µν ( x ) acts like an external source for the energy-momentum tensor T µν ( x ). We are using in (2.1)for relativistic theories the abbreviation Z x = Z d d x √ g = Z d d x q − det( g µν ( x )) . (2.2)The space-time metric g µν ( x ) has signature ( − , + , + , +).The functional integral R Dχ is defined as usual. Even though we do not write this explicitly, amicroscopic theory S [ χ ] is first defined in the presence of ultraviolet (and possibly infrared) regularizationand for renormalizable theories there is a renormalization procedure that allows to make the correspondingultraviolett scale arbitrarily large, in particular much larger than any other relevant mass scale. It isthrough this procedure that a more rigorous definition of the partition function must be done.It is worth to note here that the partition function in (2.1) depends also on the quantum state ordensity matrix ρ through the boundary conditions in the temporal domain. This is in particular importantin non-trivial situations such as at finite temperature, density, or out-of-equilibrium. For initial valueproblems one should work with the Schwinger-Keldysh double time path formalism. For the present paperwe keep the boundary condition, and therefore the state dependence, implicit. This could be easily changed,however.From the partition function one defines the Schwinger functional W [ J ] = i ln Z [ J ] and from there the quantum effective action or one-particle irreducible effective action Γ[ φ ] as a Legendre transform,Γ[ φ ] = sup J (cid:18)Z x J ( x ) φ ( x ) − W [ J ] (cid:19) . (2.3)The effective action Γ[ φ ] depends on φ , which is the expectation value of the field χ . To see this oneevaluates the supremum by varying the source field J ( x ) leading to φ ( x ) = 1 p g ( x ) δδJ ( x ) W [ J ] = 1 p g ( x ) iZ [ J ] δδJ ( x ) Z [ J ] = R Dχ χ ( x ) e iS [ χ ] − i R Jχ R Dχ e iS [ χ ] − i R Jχ . One also writes this as φ ( x ) = h χ ( x ) i , with the obvious definition of the expectation value h·i in the presence of sources J . In addition, W [ J ] andΓ[ φ ] depend on additional sources that have beed introduced for composite operators, such as the metric g µν ( x ). – 4 –n interesting property of Γ[ φ ] is its equation of motion. It follows from the variation of (2.3) as δδφ ( x ) Γ[ φ ] = p g ( x ) J ( x ) . (2.4)In particular, for vanishing source J = 0, one obtains an equation that resembles very much the classicalequation of motion, δS/δχ = 0. However, in contrast to the latter, (2.4) contains all corrections fromquantum fluctuations! Another interesting property is that tree-level Feynman diagrams become formallyexact when propagators and vertices are taken from the effective action Γ[ φ ] instead of the microscopicaction S [ χ ]. Both the microscopic action S [ χ ] and the quantum effective action Γ[ φ ] are functionals of fields. Whenone speaks of a symmetry transformation of the action one means in practise a symmetry transformationof the fields on which the action depends. A symmetry of the microscopic action means an identity of theform S [ g χ ] = S [ χ ] . (2.5)Here the group element g ∈ G is acting on the fields (not necessarily linearly) and the symmetry impliesthat the action is unmodified by this transformation. (More generally, the right hand side could differ fromthe left hand side by a constant or boundary term that does not affect the equations of motion.) So far, G could be either a finite group, an infinite discrete group or a continuous Lie group. In the latter caseone can compose finite group transformations out of infinitesimal transformations. One can write for thelatter χ ( x ) → g χ ( x ) = χ ( x ) + dχ ( x ) = χ ( x ) + idξ j ( T j χ )( x ) , (2.6)where T j is an appropriate representation of the Lie algebra acting on the fields χ (at this point notnecessarily linearly). The microscopic action transforms as S [ g χ ] = S (cid:2)(cid:0) + idξ j T j (cid:1) χ (cid:3) = S [ χ ] + Z d d x (cid:26) δS [ χ ] δχ ( x ) idξ j ( T j χ )( x ) (cid:27) , (2.7)One also abbreviate the right hand side of (2.7) as S [ χ ] + dS [ χ ] and a continuous symmetry correspondsthen to the statement dS [ χ ] = 0. The functional integral measure can also be transformed. One says that it is invariant if Dχ = D ( g χ ) . (2.8)An example for this would be a generalization of a change of integration variable x j → f j ( ~x ), where onewould in general expect a Jacobian determinant, d N x = | det( ∂x j /∂f k ) | d N f ( x ). The above equation tellsthat this determinant is unity. (More generally one may also allow a field independent constant that canbe dropped for many purposes.) Again this goes for elements g ∈ G of finite, discrete or continuous groups G . It can happen that one finds a symmetry of a microscopic action S [ χ ] but that the properly definedfunctional integral measure is not invariant. In that case one speaks of a quantum anomaly .– 5 –n the following we will use a somewhat generalized notion of anomaly which also applies to trans-formations that are not necessarily symmetries of the microscopic action but may correspond to extendedsymmetries (see below). It is a priori not clear how the functional integral measure behaves with respectto these transformations. This will be an important question to be addressed in the future. We now specialize to continuous transformations which we can study in the infinitesimal form (2.6). Aftera change of integration variable χ → g χ we write the Schwinger functional (2.1) as Z [ J ] = Z D (cid:0) χ + idξ j T j χ (cid:1) e iS [ χ + idξ j T j χ ] − i R x J ( x ) ( χ ( x )+ idξ j T j χ ( x ) ) . We now assume the invariance of the measure D (cid:0) χ + idξ j T j χ (cid:1) = Dχ , or, in other words, the absence ofan anomaly. This leads for small dξ j to Z [ J ] = Z Dχ e iS [ χ + idξ j T j χ ] − i R x J ( x ) ( χ ( x )+ idξ j T j χ ( x ) )= Z Dχ " Z x ( − p g ( x ) δδχ ( x ) S [ χ ] + J ( x ) ! dξ j T j χ ( x ) ) e iS [ χ ] − i R x J ( x ) χ ( x ) . The leading term on the right hand side is just Z [ J ] itself. Subtracting it we find using (2.4) the Slavnov-Taylor identity h dS [ χ ] i = (cid:28)Z d d x (cid:26)(cid:18) δδχ ( x ) S [ χ ] (cid:19) idξ j T j χ ( x ) (cid:27)(cid:29) = Z d d x (cid:26)(cid:18) δδφ ( x ) Γ[ φ ] (cid:19) idξ j h T j χ ( x ) i (cid:27) . (2.9)An important class of transformations is such that the Lie algebra generators T j act on the fields χ in alinear way. In that case one can write h T j χ ( x ) i = T j h χ ( x ) i = T j φ. (2.10)In that case the right hand side of (2.9) can be written as d Γ[ φ ] and one has h dS [ χ ] i = d Γ[ φ ] . (2.11)In particular, the most important case is here that the microscopic action is invariant, dS [ χ ] = 0, fromwhich it follows that also the effective action is invariant, d Γ[ φ ] = 0, orΓ[ g φ ] = Γ[ φ ] . (2.12)In summary, in the absence of a quantum anomaly, and for a linear representation of the Lie algebra onthe fields, we conclude that the effective action Γ[ φ ] shares the symmetries of the microscopic action S [ χ ].This is very useful in practice because it constrains very much the form the effective action can have. Thisis important for example for proofs of renormalizability or also for solving renormalization group equationsin practice [1]. – 6 – .5 Extended symmetries Interestingly, eq. (2.11) can also useful when the microscopic action S [ χ ] is not invariant, i. e. dS [ χ ] = 0.For example, if dS [ χ ] is linear in the field χ , one can infer that the effective action Γ[ φ ] must change undera corresponding transformation of the expectation value field φ in an analogous way such that eq. (2.11)remains fulfilled. This can also constrain the form of Γ[ φ ] substantially [40–42].When dS [ χ ] is non-linear in the fields χ , eq. (2.11) is less useful to constrain the form of Γ[ φ ], becausethe left hand side involves then expectation values of composite operators that are not easily available on thelevel of the effective action Γ[ φ ]. For example, the connected two-point correlation function h χχ i − h χ ih χ i involves the inverse of the second functional derivative of Γ[ φ ]. In such a case one may however use eq.(2.11) to calculate the expectation value on the left hand side through a simple transformation of theeffective action Γ[ φ ].Another useful case is when the left hand side of eq. (2.11) is linear in composite operators that areavailable on the level of the effective action Γ[ φ ] because external sources have been coupled to them.An example would be when dS [ χ ] involves the energy-momentum tensor which can be obtained from afunctional derivative of Γ with respect to the metric g µν ( x ).All this assumes invariance of the functional integral measure, though. One can also often make use outof anomalous symmetries, i. e. symmetry transformations where the functional measure is not invariant,if the corresponding Jacobian determinant can be absorbed into a change of the action but we will notdiscuss this further here. We now come to the second implication of symmetries besides constraints to the form of the microscopicand quantum effective action, namely conservation laws. We discuss two methods that can be used toextract conserved currents from the quantum effective action and will argue that the second method issuperior to the first. For the first method one makes the symmetry transformations space-time dependent,while for the second method one also introduces an appropriate external gauge field. Before we go intothis, let us first discuss why it is useful to obtain a Noether current from the effective action instead of themicroscopic action.
On the classical level of a field theory one can obtain the Noether currents from the microscopic action S [ χ ] through the standard textbook procedure. Sometimes such an expression is useful also for a quantumdescription as a starting point to calculate its expectation value when quantum and statistical fluctuationsare taken into account. Alternatively one can obtain the expectation value of a Noether current directlyfrom the quantum effective action as will be discussed below.However, one should be aware of the fact that the difference between the classical or microscopicaction and the quantum effective action is a rather non-trivial result of quantum and possibly statisticalfluctuations. In particular, quantum fluctuations are present on all scales and need proper regularizationand renormalization procedures. It is well known that quantum fluctuations can modify a theory substan-tially, for example the propagating degrees of freedom can change from fundamental fields to composite– 7 –elds for bound states [58–60]. In this sense it is possible that Noether currents differ substantially inform when derived from microscopic actions or quantum effective actions, respectively. There are powerfulnon-perturbative theoretical techniques to calculate the quantum effective action, such as the functionalrenormalization group. It allows to take quantum fluctuations step-by-step into account and to find suitableapproximations at each scale.The above arguments show that it can be rather advantageous to calculate Noether currents directlyfrom the quantum effective action, instead of aiming at an expression in terms of the microscopic actionand expectation values in terms of complicated composite operators.One should note here, however, that Noether currents that follow from the quantum effective actionare usually not monomials or polynomials in the field expectation values and their derivatives. There canbe additional terms that do not vanish, even when the field expectation values do. For example, at finitetemperature, the energy-momentum tensor is non-zero even when the field expectation values vanish. Also,one should keep in mind that the effective action is state-dependent and accordingly the Noether currentsderived from it also are. Again, the finite temperature state may serve as an example. Consider the transformation of the fields φ ( x ) → φ ( x ) + idξ j ( x ) T j φ ( x ) , (3.1)where dξ j ( x ) are space-time position-dependent, infinitesimal parameters. The generators T j act linearlyon the field expectation value fields φ ( x ) such that eq. (2.10) holds. The transformation law (3.1) isinherited from the corresponding transformation of the microscopic fields in eq. (2.6).The quantum effective action Γ[ φ ] changes to linear order in dξ j like [61]Γ[ φ + idξ j T j φ ] = Γ[ φ ] + Z d d x √ g (cid:26) I j ( x ) dξ j ( x ) + J µj ( x ) ∇ µ dξ j ( x ) + 12 K µνj ( x ) ∇ µ ∇ ν dξ j ( x ) + . . . (cid:27) . (3.2)If the transformation (3.1) is a global symmetry of the effective action, this implies that I j ( x ) = 0 so thatthe expansion on the right hand side of (3.2) starts with the second term which has one derivative. However,one can also consider transformations that are not global symmetries such that I j ( x ) is non-vanishing. Onecan now write eq. (3.2) after partial integration as Z d d x √ g dξ j ( x ) (cid:26) √ g δ Γ δφ ( x ) iT j φ ( x ) − I j ( x ) + ∇ µ J µj ( x ) − ∇ µ ∇ ν K µνj ( x ) + . . . (cid:27) = 0 . (3.3)Surface terms have been dropped here. Because dξ j ( x ) is arbitrary, this implies1 √ g δ Γ δφ ( x ) iT j φ ( x ) − I j ( x ) + ∇ µ J µj ( x ) − ∇ µ ∇ ν K µνj ( x ) + . . . = 0 . (3.4)Using the field equation (2.4) one obtains a set of conservation-type relations ∇ µ (cid:18) −J µj ( x ) + 12 ∇ ν K µνj ( x ) − . . . (cid:19) = iJ ( x ) T j φ ( x ) − I j ( x ) . (3.5)For a transformation which defines a global symmetry such that I j ( x ) = 0 and for vanishing source J = 0the right hand side vanishes and the expression in brackets on the left hand side of (3.5) defines then a setof conserved Noether currents. – 8 –et us note here that a relation of the type (3.5) as derived from a quantum effective action mayalso be useful when I j ( x ) is non-vanishing, as long as this field is known, for example because an externalsource is coupled to it. In that case one may call the expression in brackets on the left hand side of (3.5)a set of non-conserved Noether currents corresponding to an extended symmetry .A problem with the above derivation is that the expansion on the left hand side of (3.5) does ingeneral not terminate. This makes the entire construction somewhat implicit. A notable exception is whenΓ[ φ ] = S [ φ ] is the microscopic or classical action which contains at most second derivatives of the fieldssuch that the equations of motion are partial differential equations of at most second order. In this casethe above construction leads to the standard Noether currents of the classical theory. In contrast, whenthe effective action Γ[ φ ] differs from S [ φ ] by the effect of quantum fluctuations, one can not assume thatonly low orders of a derivatives of the fields are present. In fact, the quantum effective action contains ingeneral all orders of a derivative expansion, as well as non-perturbative terms.These remarks show that an alternative approach is needed to obtain Noether currents from thequantum effective action when the latter cannot be assumed to have a derivative expansion terminating atfinite order. To such a construction we turn next. Let us now introduce an external gauge field for the local transformation (3.1). All derivatives of the field χ ( x ) in the microscopic action and of the expectation value field φ ( x ) in the quantum effective action Γ[ φ ]are now replaced by covariant derivatives, D µ φ ( x ) = (cid:0) ∇ µ − iA jµ ( x ) T j (cid:1) φ ( x ) . (3.6)When the generators T j are not commuting, the external gauge field is non-abelian. In that case we mayintroduce structure constants through the relation[ T k , T l ] = if jkl T j . (3.7)The transformations of the expectation value fields φ ( x ) continue to be of the form (3.1), while the(non-abelian) gauge fields transforms as usual according to A jµ ( x ) → A jµ ( x ) + f jkl A kµ ( x ) dξ l ( x ) + ∇ µ dξ j ( x ) = A jµ ( x ) + ( D µ dξ ) j . (3.8)In the last equation we defined the covariant derivative of dξ j in a variant of the adjoint representation.We will call an infinitesimal transformation dξ j ( x ) “global” when ( D µ dξ ) j ( x ) = 0.We assume now for simplicity that this gauged transformation is free from any anomalies, so that theanalysis in section 2 goes through for this transformation, as well. In particular, when the microscopicaction S [ χ, A ] has a symmetry or extended symmetry under the transformation (3.8), this will also be thecase for the quantum effective action Γ[ φ, A ]. An important consequence is that derivatives of fields φ can appear in Γ[ φ, A ] only as covariant derivatives of the form (3.6). One should note here, however, thatΓ[ φ, A ] can contain covariant derivatives of arbitrary order, and also non-local gauge invariant terms.The gauge field has been introduced in such a way that a local transformation (with space-time-dependent dξ j ( x )) is in fact equivalent to a global transformation when A jµ ( x ) is transformed, as well. In– 9 –ther words, the only effect of a locally varying dξ j ( x ) is taken up by the transformation behavior of A jµ ( x ).One has now for the transformation of the effective actionΓ[ φ + idξ j T j φ, A jµ + f jkl A kµ dξ l + ∇ µ dξ j ] = Γ[ φ ] + Z d d x √ g (cid:8) I j ( x ) dξ j ( x ) (cid:9) . (3.9)In contrast to (3.2) no higher order derivatives are present on the right hand side of (3.9). This is aconsequence of the external gauge field, and the fact that all derivatives have been replaced by covariantderivatives (3.6). Note that this includes possible regulator terms that have been added to make thefunctional integral well defined.We can write now Z d d x √ g ((cid:18) √ g δ Γ δφ ( x ) iT j φ ( x ) − I j ( x ) (cid:19) dξ j ( x ) + 1 √ g δ Γ δA jµ ( x ) (cid:16) f jkl A kµ ( x ) dξ l ( x ) + ∇ µ dξ j ( x ) (cid:17)) = 0 . (3.10)Using partial integration, the field equation (2.4), and the fact that dξ j ( x ) is arbitrary, this implies nowwith the definition J µj ( x ) = 1 √ g δ Γ δA jµ ( x ) , (3.11)the covariant conservation-type relation D µ J µj ( x ) = ∇ µ J µj ( x ) + f ljk A kµ ( x ) J µl ( x ) = iJ ( x ) T j φ ( x ) − I j ( x ) . (3.12)The first equation defines a covariant derivative D µ for the current J µj ( x ) in a variant of the adjointrepresentation of the gauge symmetry.If there is a global symmetry, i. e. if the effective action is invariant for ( D µ dξ ) j = 0, one has I j ( x ) = 0and for vanishing external sources J ( x ) = A jµ ( x ) = 0, this is indeed a covariant conservation law of thestandard form ∇ µ J µj = 0. More general, the “source term” on the right hand side of (3.12) might benon-vanishing. The equation is then still potentially very useful, as long as the source term is knownexplicitly.In summary, the above discussion gives a recipe to derive conservation-type equations by introducinggauge fields associated with fields transformations in the partition function and taking functional derivativeswith respect to it. When the transformation corresponds to a real symmetry one obtains in this way aconserved Noether current for which all quantum corrections have been taken into account. When thetransformation is instead an extended symmetry one obtains a conservation-type equation with a sourceterm on the right hand side. It should also be seen as a macroscopic equation of motion for which quantumcorrections have been taken into account. In the following we discuss a number of transformations related to space-time geometry. Following thegeneral principles introduced in section 3.3 we introduce appropriate (external) gauge fields and discusswhat kind of conservation laws follow from their variation. We start our discussion with general coordinatetransformations which can be seen as a localized version of translations. Subsequently we discuss localchanges of frame in the tangent space of the space-time manifold. This will be done first for the restricted– 10 –et of orthonormal frames where the transformations can be seen as local versions of Lorentz transfor-mations. Subsequently we turn to general linear local frame changes which include besides local Lorentztransformations also local dilatations as well as shear transformations. We argue that the latter two shouldbe understood as extended symmetries in general. In all cases we discuss what are the associated conservedand non-conserved Noether currents.
We start with a one-particle irreducible or quantum effective action Γ[ φ, g ] that depends besides the fieldexpectation values φ ( x ) on the space-time metric g µν . For the present subsection we assume that φ ( x )contains only fields of integer spin or, in other words, that fermionic fields have been fully integrated outfrom the partition function at vanishing source. This simplifies somewhat the discussion in the sense thatwe do not yet have to introduce a tetrad. The matter fields φ ( x ) could be scalars, vectors or tensors withrespect to general coordinate transformations.Under a general coordinate transformation or diffeomorphism x µ → x ′ µ ( x ), the metric transforms like g µν ( x ) → g ′ µν ( x ′ ) = ∂x ρ ∂x ′ µ ∂x σ ∂x ′ ν g ρσ ( x ) . (4.1)Changing afterwards the coordinate label from x ′ µ back to x µ gives the transformation rule g µν ( x ) → g ′ µν ( x ) = ∂x ρ ∂x ′ µ ∂x σ ∂x ′ ν g ρσ ( x ) − (cid:2) g ′ µν ( x ′ ) − g ′ µν ( x ) (cid:3) . (4.2)For an infinitesimal transformation x ′ µ = x µ − ε µ ( x ) this reads g µν ( x ) → g µ ( x ) + ε ρ ( x ) ∂ ρ g µν ( x ) + ( ∂ µ ε ρ ( x )) g ρν ( x ) + ( ∂ ν ε ρ ( x )) g µρ ( x )= g µν ( x ) + L ε g µν ( x ) . (4.3)We are using here the Lie derivative L ε in the direction ε µ ( x ). More general, any coordinate tensor fieldtransforms under infinitesimal general coordinate transformations with the corresponding Lie derivative L ε . This fixes in particular how the components of the matter fields φ ( x ) transform.In the following we will also need the covariant derivative. In the present section it is based on theLevi-Civita connection given by the Christoffel symbols of second kind,Γ ρµ ν = (cid:26) ρµν (cid:27) = 12 g ρλ ( ∂ µ g νλ + ∂ ν g µλ − ∂ λ g µν ) . (4.4)For future reference we note also the variation of the Levi-Civita connection, which can be written as δ Γ ρµ ν = 12 g ρλ ( ∇ µ δg νλ + ∇ ν δg µλ − ∇ λ δg µν ) . (4.5)Note that in contrast to the Christoffel symbol Γ ρµ ν itself, its variation δ Γ ρµ ν is actually a coordinatetensor.The change of the metric in eq. (4.3) can also be written as g µν ( x ) → g µν ( x ) + ∇ µ ε ν ( x ) + ∇ ν ε µ ( x ) . (4.6)– 11 –his illustrates that the metric can be seen here as the gauge field of general coordinate transformations.Variation of the effective action Γ[ φ, g ] with respect to the metric at stationary matter fields, δ Γ /δφ = 0,yields the energy-momentum tensor, δ Γ[ φ, g ] = 12 Z d d x √ g T µν ( x ) δg µν ( x ) . (4.7)In fact, T µν ( x ) as defined by this expression should be seen as the expectation value of the symmetric energy-momentum tensor (see also below) in the state that defines the effective action Γ[ φ, g ]. The variationincludes the connection, with (4.5) obeyed.Inserting (4.3) in (4.7) shows that invariance under general coordinate transformations yields thecovariant conservation law for the energy momentum tensor ∇ µ T µν ( x ) = 0 . (4.8)In this sense, one may see the covariant conservation of energy and momentum as a special case of thegeneral principles discussed in section 3.3. The Levi-Civita connection is uniquely determined by being both metric compatible and torsion free. Itseems that the space-time we inhabit fulfills these two conditions to an excellent approximation. Never-theless, it is interesting to relax these constraints and to study more general connections. Usually this isdone in order to understand and constrain alternative theories of gravitation in more detail. For us thepurpose is different: We are interested in constraining the form of the effective action for matter fields andto derive conservation-type relations. A very interesting possibility to this end is to study the quantumfield theory in a geometry characterized by a general affine connection and to take functional derivativesof the quantum effective action with respect to the connection field.
Parallel transport.
As a starting point for the definition of a covariant derivative one may take thenotion of a parallel transport. The rule is here that a vector field U µ ( x ) counts as parallelly displaced froma position x µ to x µ + dx µ when it changes by dU ρ ( x ) = − (cid:2) Γ ρµ σ ( x ) − ∆ U B µ ( x ) δ ρσ (cid:3) U σ ( x ) dx µ . (4.9)The square bracket on the right hand side contains two terms. The first is a geometric part proportional tothe affine connection Γ ρµ σ ( x ) which generalizes the Levi-Civita connection. For the second term we takethe field U ρ to have the (momentum or mass) scaling dimension or conformal weight ∆ U . The Weyl gaugefield B µ ( x ) performs an additional local scaling of the field U ρ ( x ). In contrast to the first term in (4.9),the second term or dilatation term is also present for scalar fields ϕ ( x ) when they have a non-vanishingscaling dimension ∆ ϕ and when the Weyl gauge field B µ ( x ) is non-vanishing. Co-covariant derivative.
The so-called co-covariant derivative [62, 63] associated to the parallel trans-port (4.9) is given by ∇ µ U ρ ( x ) = ∂ µ U ρ ( x ) + (cid:2) Γ ρµ σ ( x ) − ∆ U B µ ( x ) δ ρσ (cid:3) U σ ( x ) . (4.10)– 12 –n particular this vanishes when U ρ ( x ) is parallelly transported according to (4.9). Eq. (4.10) is easilygeneralized to other tensor fields in a coordinate basis. For example, the co-covariant derivative of a tensorfield χ ρλ ( x ) with scaling dimension ∆ χ would be ∇ µ χ ρλ ( x ) = ∂ µ χ ρλ ( x ) + Γ ρµ σ ( x ) χ σλ ( x ) − Γ τµ λ ( x ) χ ρτ ( x ) − ∆ χ B µ ( x ) χ ρλ ( x ) . (4.11)For a scalar field ϕ ( x ) the co-covariant derivative is given by ∇ µ ϕ ( x ) = ∂ µ ϕ ( x ) − ∆ ϕ B µ ( x ) ϕ ( x ) . (4.12)The co-covariant derivative has its name because it is covariant with respect to both general coordinatetransformations x → x ′ ( x ) and local scaling or Weyl gauge transformations φ ( x ) → e − ∆ φ ζ ( x ) φ ( x ) , g µν ( x ) → e ζ ( x ) g µν ( x ) . (4.13) Affine connection.
Generalizing beyond the Levi-Civita connection (4.4) one may write the affine con-nection as Γ ρµ σ = (cid:26) ρµσ (cid:27) + N ρµ σ = 12 g ρλ ( ∂ µ g σλ + ∂ σ g µλ − ∂ λ g µσ ) + N ρµ σ , (4.14)where N ρµ σ is known as the deviation or distortion tensor. (It transforms indeed as a tensor under generalcoordinate transformations, in contrast to Γ ρµ σ .) Co-covariant and Levi-Civita covariant derivatives.
We will use in the following a notation where ∇ µ denotes the co-covariant derivative as introduced in (4.10), while ∇ µ is the ordinary covariant derivativebased on the Levi-Civita connection (4.4). Equation (4.10) can also be written as ∇ µ U ρ ( x ) = ∇ µ U ρ ( x ) + (cid:2) N ρµ σ ( x ) − ∆ U B µ ( x ) δ ρσ (cid:3) U σ ( x ) . (4.15) Non-metricity.
The co-covariant derivative of the metric itself is given by ∇ µ g ρσ ( x ) = − [ N µρσ ( x ) + N µσρ ( x )] − ∆ g B µ ( x ) g ρσ ( x ) . = − [ N µρσ ( x ) + N µσρ ( x )] + 2 B µ ( x ) g ρσ ( x ) . (4.16)In the second line we have used that the metric g µν ( x ) has conformal weight ∆ g = −
2, as follows from eq.(4.13). The first term on the right hand side of (4.16), namely the combination B µρσ ( x ) = 12 [ N µρσ ( x ) + N µσρ ( x )] = − ∇ µ g ρσ ( x ) (4.17)is known as the non-metricity tensor. It is obviously symmetric in the last two indices. (Our conventiondiffers by the factor 1 / B ρµ σ ( x ) = ˆ B ρµ σ ( x ) + B µ ( x ) δ ρσ , (4.18)where ˆ B ρµ σ ( x ) is trace-less and sometimes called proper non-metricity tensor, ˆ B ρµ ρ ( x ) = 0, and B µ ( x ) =(1 /d ) B ρµ ρ ( x ) corresponds to the trace of the non-metricity tensor and is the Weyl vector or Weyl gauge field introduced already in eq. (4.9).Note that the full co-covariant derivative of the metric in (4.16) is in fact given by the proper non-metricity tensor − B µρσ ( x ). – 13 – orsion. Consider the commutator of two co-covariant derivatives acting on a scalar field ϕ ( x ), ∇ µ ∇ ν ϕ ( x ) − ∇ ν ∇ µ ϕ ( x ) = − T ρµν ( x ) ∇ ρ ϕ ( x ) − ∆ ϕ [ ∂ µ B ν ( x ) − ∂ ν B µ ( x )] ϕ ( x ) . (4.19)This contains two kinds of field strengths. One is the torsion tensor which is formally defined through thefollowing combination of vector fields with vanishing scaling dimension, T ( U, V ) = ∇ U V − ∇ V U − [ U, V ].In components it is given by the anti-symmetric part of the affine connection, T ρµσ ( x ) = Γ ρµ σ ( x ) − Γ ρσ µ ( x ) = N ρµ σ ( x ) − N ρσ µ ( x ) . (4.20)The second term in (4.19) is the combination B µν ( x ) = ∂ µ B ν ( x ) − ∂ ν B µ ( x ) known as the segmentalcurvature tensor (see also below). Decomposition of distortion tensor.
Using eqs. (4.17) and (4.20) we may write the distortion tensoras N ρµ σ = 12 (cid:2) T ρµ σ − T ρσµ + T ρµσ (cid:3) + B ρµ σ + B ρσµ − B ρµσ = C ρµ σ + D ρµ σ . (4.21)The combination C ρµ σ = 12 (cid:2) T ρµ σ − T ρσµ + T ρµσ (cid:3) (4.22)is known as the contorsion tensor . It is anti-symmetric in the last two indices, C µρσ = − C µσρ , so it doesnot contribute to the non-metricity in (4.17). In contrast, the combination D ρµ σ = B ρµ σ + B ρσµ − B ρµσ = ˆ B ρµ σ + ˆ B ρσµ − ˆ B ρµσ + B µ δ ρσ + B σ δ ρµ − B ρ g µσ (4.23)which we may call con-metricity tensor , is symmetric in µ and σ and does not contribute to torsion in eq.(4.20).We may therefore write the torsion tensor in terms of contorsion as T ρµσ ( x ) = C ρµ σ ( x ) − C ρσ µ ( x ) , (4.24)and the non-metricity tensor in terms of the con-metricity tensor as B µρσ ( x ) = 12 [ D µρσ ( x ) + D µσρ ( x )] . (4.25)The proper non-metricity ˆ B µρσ ( x ) corresponds to the symmetric and trace-less part of con-metricity withrespect to the last two indices. However, con-metricity has also an anti-symmetric part. The Weyl gaugefield can also be obtained directly from the trace of con-metricity as B µ ( x ) = 1 d D ρµ ρ ( x ) . (4.26)With this, the trace of the complete affine connection (4.14) can be written asΓ ρµ ρ ( x ) = 1 p g ( x ) ∂ µ p g ( x ) + d B µ ( x ) . (4.27)– 14 –n this sense the Weyl gauge field is actually determined by the affine connection and the metric, B µ ( x ) = 1 d " Γ ρµ ρ ( x ) − p g ( x ) ∂ µ p g ( x ) . (4.28)For our purposes it is particularly useful to work with contorsion C ρµ σ , the Weyl gauge field B µ andproper non-metricity ˆ B ρµ σ as the fields that parametrize the distortion tensor, so that the full connectionbecomes Γ ρµ σ = 12 g ρλ ( ∂ µ g σλ + ∂ σ g µλ − ∂ λ g µσ ) + C ρµ σ + ˆ B ρµ σ + ˆ B ρσµ − ˆ B ρµσ + B µ δ ρσ + B σ δ ρµ − B ρ g µσ . (4.29) Variation of affine connection.
The full variation of the affine connection is now given by δ Γ ρµ σ = 12 g ρλ ( ∇ µ δg σλ + ∇ σ δg µλ − ∇ λ δg µσ ) + δC ρµ σ + δD ρµ σ . (4.30)The covariant derivative on the right hand side uses the Levi-Civita connection. In particular it followsfrom (4.30) that all components of the connection field Γ ρµ σ can be varied free of constraints when thisvariation is understood as a superposition of the variation of the Christoffel symbols due to a variationof the metric and variations of the torsion tensor and non-metricity tensor. In some situations one mayfurther restrict this and demand for example that the non-metricity vanishes. Curvature tensor.
One may define the curvature tensor by the commutator of covariant derivatives ofvector fields with vanishing scaling dimension, R ( U, V ) W = ∇ U ∇ V W − ∇ V ∇ U W − ∇ [ U,V ] W. (4.31)In components, R ρσµν = ∂ µ Γ ρν σ − ∂ ν Γ ρµ σ + Γ ρµ λ Γ λν σ − Γ ρν λ Γ λµ σ = R ρσµν + ∇ µ N ρν σ − ∇ ν N ρµ σ + N ρµ λ N λν σ − N ρν λ N λµ σ . (4.32)This is obviously anti-symmetric in the last two indices. In the second line of (4.32), R ρσµν is the standardRiemann tensor based on the Levi-Civita connection and the covariant derivatives are also based on theLevi-Civita connection.It is also useful to have the variation of (4.32) at hand. It can be written as δR ρσµν = ∇ µ δ Γ ρν σ − ∇ ν δ Γ ρµ σ + T λµν δ Γ ρλ σ , (4.33)with torsion as in (4.24). We are using here the co-covariant derivative with vanishing scaling dimensionfor the variation of the connection δ Γ ρν σ . Ricci scalar.
There is a unique complete contraction of (4.32) which forms the analog of the Ricci scalar R = R ρσρσ , R = R ρσρσ = R + ∇ ρ N ρσσ − ∇ σ N ρσρ + N ρρ λ N λσσ − N ρσ λ N λσρ = R + 2 ∇ ρ C ρσσ + 2 ∇ ρ ˆ B σρσ − d − ∇ ρ B ρ + N ρρ λ N λσσ − N ρσ λ N λσρ . (4.34)– 15 –et us note that in the presence of torsion one can alternatively define a version of the curvature tensoron the transposed affine connection ˘Γ ρµ σ = Γ ρµ σ − T ρµσ = Γ ρσ µ , but we will not discuss this further. Also,in contrast to Riemann geometry, R ρσµν is in general not anti-symmetric in the first two indices. Basedon (4.32) one may define different contractions. One is the segmental curvature tensor B µν = 1 d R ρρµν = 1 d (cid:2) ∂ µ Γ ρν ρ − ∂ ν Γ ρµ ρ (cid:3) = ∂ µ B ν − ∂ ν B µ . (4.35)The other two possibilities are R µν = R ρµρν and R ρµ νρ which both equal the standard Ricci tensor in theabsence of non-metricity and torsion.The variation of the Ricci scalar is given by δR = − R µν δg µν + g νσ ∇ ρ δ Γ ρν σ − g νσ ∇ ν δ Γ ρρ σ + T λ σρ δ Γ ρλ σ . (4.36)On the right hand side one may use (4.30) for further simplifcation. In the absence of contorsion andnon-metricity this reduced to the standard identity δR = − R µν δg µν + [ ∇ µ ∇ ν δg µν − g µν ∇ ρ ∇ ρ δg µν ] . (4.37) In the following we will investigate how a quantum effective action for matter fields reacts the contor-sion and non-metrcity as external sources and specifically what kind of equations can be derived fromtransformations for which the connection acts as a gauge field.Let us write the variation of the action with respect to the metric g µν and the connection Γ ρµ σ as δ Γ = Z d d x √ g (cid:26) U µν ( x ) δg µν ( x ) − S µ σρ ( x ) δ Γ ρµ σ ( x ) (cid:27) . (4.38)The variation with respect to g µν ( x ) at fixed connection defines a symmetric tensor U µν ( x ), while thevariation with respect to the connection at fixed metric defines a tensor field S µ σρ ( x ). The latter is knownas hypermomentum current [43, 44, 46, 47]. We have assumed in (4.38) that the Weyl gauge field B µ ( x )has been expressed through eq. (4.28) in terms of the affine connection and the metric.It is conventional and convenient to further decompose the hypermomentum current S µ σρ ( x ) accordingto S µ σρ ( x ) = Q µ σρ ( x ) + W µ ( x ) δ σρ + S µ σρ ( x ) + S σµρ ( x ) + S µσρ ( x ) . (4.39)Here S µ σρ ( x ) is anti-symmetric, S µρσ ( x ) = − S µσρ ( x ), in the last two indices and known as the spin current .It can be written in terms of the hypermomentum as S µρσ = 12 ( S µρσ − S µσρ ) . (4.40)In contrast, Q µ σρ ( x ) is symmetric in the last two indices, Q µρσ ( x ) = Q µσρ ( x ), and traceless, Q µ ρρ ( x ) = 0,and known as the (intrinsic) shear current . Finally, W µ ( x ) is the (intrinsic) dilatation current or Weylcurrent. With these definitions we follow ref. [43–45]. The combination of shear current and dilatationcurrent can be written in terms of the hypermomentum current as Q µρσ + W µ g ρσ = 12 ( S µρσ + S ρσµ − S ρµσ + S µσρ + S σρµ − S σµρ ) . (4.41)– 16 –sing eqs. (4.30), (4.18) and (4.39) one can write (4.38) as δ Γ = Z d d x √ g ( U µν δg µν − S µρσ ( ∇ µ δg σρ + ∇ σ δg µρ − ∇ ρ δg µσ ) − S µ σρ δC ρµ σ − Q µ σρ δ ˆ B ρµ σ − d W µ δB µ ) . (4.42)The first line gives the full variation of the effective action with respect to the metric at fixed (typicallyvanishing) distortion tensor. Partial integration and the use of δg µν = δg νµ allows to write this part as δ Γ = 12 Z d d x √ g (cid:20) U µν + 14 ∇ ρ ( S ρµν + S ρνµ + S µνρ + S νµρ − S µρν − S νρµ ) (cid:21) δg µν . (4.43)Because this must equal δ Γ = 12 Z d d x √ g T µν δg µν , (4.44)we find for the energy-momentum tensor the decomposition T µν = U µν + 14 ∇ ρ ( S ρµν + S ρνµ + S µνρ + S νµρ − S µρν − S νρµ )= U µν + 12 ∇ ρ ( Q ρµν + W ρ g µν ) . (4.45)Let us note here that U µν ( x ) is in general not conserved by itself. The contributions from the shear currentand dilatation current in (4.45) are needed to obtain a conservation law. However, these two terms comeunavoidably with derivatives.Taking the trace of (4.45) we obtain the divergence-type relation ∇ ρ W ρ = 2 d ( T µµ − U µµ ) . (4.46)Similarly, by subtracting the trace we find ∇ ρ Q ρµν = 2( T µν − U µν ) − d ( T σσ − U σσ ) g µν . (4.47)We will argue below that these two relations should be understood as conservation-type relations for non-conserved Noether currents associated to extended symmetries. While variants of eq. (4.46) have beendiscussed in the context of dilatation and conformal symmetry (see also section 4.4 below), eq. (4.47) isnew to the best or our knowledge.The decomposition in (4.45) is particularly interesting from the point of view of relativistic fluiddynamics and its derivation from quantum field theory. The first part contains the equilibrium part of theenergy-momentum tensor, while the second term is by construction at least one order higher in derivativesand can give a non-equilibrium part of the energy-momentum tensor.In the following we will investigate different transformations in the frame bundle for which the affineconnection acts as a gauge field, in more detail. This will lead to further insights into the physics significanceof the spin current, dilatation current and shear current. We start with Weyl transformations and localLorentz transformations and turn then to shear transformations before we combine everything into generallinear transformations. – 17 – .4 Weyl gauge transformations It is interesting at this point to discuss dilatations or Weyl gauge transformations in more detail. Thetransformations of the metric and matter fields in (4.13) get supplemented by a transformation of theWeyl gauge field so that the complete transformation is φ ( x ) → e − ∆ φ ζ ( x ) φ ( x ) , g µν ( x ) → e ζ ( x ) g µν ( x ) , B µ ( x ) → B µ ( x ) − ∂ µ ζ ( x ) . (4.48)Interestingly, the general connection in (4.29) is left unchanged by this transformation because contributionsfrom the Levi-Civita part and the Weyl non-metricity part cancel.Let us also note here that √ g → e dζ √ g and T µν → e − (2+ d ) ζ T µν so that the energy-momentum tensorwith two upper indices has the scaling dimension ∆ T µν = 2 + d .For the effective action we find from (4.38) the following change under an infinitesimal Weyl transfor-mation δ Γ = Z d d x √ g U µν ( x ) δg µν ( x ) = Z d d x √ g U µµ ( x ) δζ ( x ) . (4.49)For a generic quantum field theory U µµ ( x ) is non-vanishing and the effective action is not invariant underWeyl transformations. An exception is a scale-invariant theory at a renormalization group fixed pointwhere U µµ ( x ) = 0. (Even then there are corrections to the right hand side in curved space due to theconformal anomaly.)It is interesting to note that a symmetry under dilatations implies U µµ ( x ) = 0 and not directly avanishing trace of the energy-momentum tensor T µµ ( x ) = 0. The latter condition would be implied bya symmetry of the theory under the larger group of conformal transformations (again up to anomalouscorrections arising in curved space).Assume now that we consider a theory that is invariant under scaling transformations so that U µµ ( x ) =0. Equation (4.46) implies then T µµ − d ∇ µ W µ = 0 , (4.50)In this context, V µ = − d W µ is known as the virial current [64]. It was shown in ref. [65] that under thecondition that the virial current is itself a divergence, V µ = ∇ ρ σ ρµ , one can actually define an “improved”energy-momentum tensor which is then trace-less. In practise this improvement can be done by changingthe way the theory couples to space-time curvature, more specifically the Ricci scalar and Ricci tensor. Infact, it has been shown [66] that if the theory has a conformal symmetry in flat space one can couple itthe Ricci tensor in such as way that the energy-momentum tensor following through eq. (4.7) is in factthe “improved” energy-momentum tensor. Assuming now that our theory is conformal and that this kindof improvement has been done implies that the energy-momentum tensor is trace-less (in flat space), andfrom (4.50) it follows that in this case ∇ µ W µ = 0 . (4.51)To summarize, the (intrinsic) dilatation or Weyl current W µ is in general not conserved and fulfills thedivergence-type relation (4.46). Because the right hand side is known (or calculable) one should understand W µ as a non-conserved Noether current. For a scale-invariant system the divergence-type relation simplifiesto (4.50). Finally, for conformal systems one has an actual conservation law for the Weyl current (4.51).One should mention here, however, that oftentimes for a conformal field theory the Weyl current actuallysimply vanishes, W µ = 0. – 18 –inally, let us mention that the conservation law associated to full dilatation symmetry in Minkowskispace (for a review see ref. [67]) has an additional part due to the scaling of coordinates. It can be writtenas J µD = x ν T µν − d W µ , (4.52)and is indeed conserved when eq. (4.50) is fulfilled. As a next step we want to investigate local changes of frame that leave the space-time metric invari-ant. One can also understand them as a local version of Lorentz transformations. Mathematically, thesetransformations correspond to changes of basis in the frame bundle restricted to orthonormal frames.Orthonormal frames are anyway needed to describe fermionic fields because the standard version ofthe Clifford algebra uses them. (For an alternative approach see ref. [68] and references therein.) A choiceof frame is usually parametrized in terms of the tetrad field, through a formalism we recall below. Thetetrad can be defined formally as a Lorentz vector valued one-form V Aµ ( x ) dx µ . The latin index A is here aLorentz index (in a sense to be made more precise below), while the Greek index µ is a standard coordinateindex. The tetrad parametrizes the change of basis in the frame bundle, and its associate bundle, fromthe holonomic or coordinate frame to an orthonormal frame. More precisely, θ A ( x ) = V Aµ ( x ) dx µ could beseen as a new basis for one-forms, out of which any one-form can be composed, ω ( x ) = ω A ( x ) θ A ( x ).We also introduce the inverse tetrad V µA ( x ) such that V Aµ ( x ) V νA ( x ) = δ νµ , V Aµ ( x ) V µB ( x ) = δ AB . (4.53)The inverse tetrad can be seen as constituting a new basis for vectors, v A ( x ) = V µA ( x ) ∂ µ such that anyvector field can be written locally as U ( x ) = U A ( x ) v A ( x ). The dual basis for one-forms is precisely θ A ( x ).With Minkowski metric η AB = diag( − , +1 , +1 , +1) one can write the coordinate metric g µν ( x ) as g µν ( x ) = η AB V Aµ ( x ) V Bν ( x ) . (4.54)Under a coordinate transformation or diffeomorphism x µ → x ′ µ ( x ) on the coordinate side, the tetradtransforms like a one-form V Aµ ( x ) → V ′ Aµ ( x ′ ) = ∂x ν ∂x ′ µ V Aν ( x ) . (4.55)Changing afterwards the label or integration variable from x ′ µ back to x µ gives the transformation rule V Aµ ( x ) → V ′ Aµ ( x ) = ∂x ν ∂x ′ µ V Aν ( x ) − (cid:2) V ′ Aµ ( x ′ ) − V ′ Aµ ( x ) (cid:3) . (4.56)For an infinitesimal transformation x ′ µ = x ′ µ − ε µ ( x ) this reads V Aµ ( x ) → V Aµ ( x ) + ε ν ( x ) ∂ ν V Aµ ( x ) + ( ∂ µ ε ρ ( x )) V Aρ ( x ) = V Aµ ( x ) + L ε V Aµ ( x ) . (4.57)We are using here again the Lie derivative L ε in the direction ε µ ( x ).From (4.54) and (4.13) one finds that under a Weyl transformation one has V Aµ ( x ) → e ζ ( x ) V Aµ ( x ) , (4.58) We are following here the conventions of ref. [69]. Other authors refer to V µA ( x ) as the tetrad field. – 19 –o that the tetrad has the scaling dimension ∆ V = −
1. (Obviously the inverse tetrad has the oppositescaling dimension.)In addition to coordinate and Weyl transformations one may also consider local Lorentz transforma-tions or changes of the orthonormal frame acting on the tetrad according to V Aµ ( x ) → V ′ Aµ ( x ) = Λ AB ( x ) V Bµ ( x ) , (4.59)where Λ AB ( x ) is at every point x a Lorentz transformation matrix such thatΛ AB ( x )Λ CD ( x ) η AC = η BD . (4.60)In other words, at every space-time point x the matrices Λ AB ( x ) are elements of the group SO(1 , d − internal , i. e. they do not act on the space-time argument x of a field as a conventional Lorentz transformation would do. In infinitesimal form, thelocal Lorentz transformation (4.59) reads V Aµ ( x ) → V ′ Aµ ( x ) = V Aµ ( x ) + dω AB ( x ) V Bµ ( x ) , (4.61)where dω AB ( x ) = − dω BA ( x ) is anti-symmetric and infinitesimal.Coordinate vector and tensor fields can be transformed using the tetrad and its inverse to becomescalars under general coordinate transformations, e. g. ϕ B ( x ) = V Bµ ( x ) ϕ µ ( x ) , χ AB ( x ) = V Aµ ( x ) V Bν ( x ) χ µν ( x ) . (4.62)The results are then Lorentz vectors and tensors, respectively. In other words these objects have nowbeen fully transformed to the orthonormal frame. At this point it is worth to note that an action that isstationary with respect to coordinate tensor fields like χ µν ( x ) is also stationary with respect to the resultingLorentz tensor field χ AB ( x ).More generally, a field Ψ might transform in some representation R with respect to the local, internalLorentz transformations or changes of orthonormal frame,Ψ( x ) → Ψ ′ ( x ) = L R (Λ( x ))Ψ( x ) , (4.63)or infinitesimally Ψ( x ) → Ψ ′ ( x ) = Ψ( x ) + i dω AB ( x ) M AB R Ψ( x ) . (4.64)One would also like to have a covariant derivative with respect to the local Lorentz transformations.This leads to the spin connection . The spin covariant derivative D µ is defined such that for the spinor fieldΨ( x ) transforming under local Lorentz transformations according to (4.63) one has V µA ( x ) D µ Ψ( x ) → Λ BA ( x ) V µB ( x ) L R (Λ( x )) D µ Ψ( x ) . (4.65)In other words, the covariant derivative of some field transforms as before, with an additional transformationmatrix for the new index, but without any extra non-homogeneous term. The full covariant derivative isnow D µ = ∇ µ + Ω µ ( x ) . (4.66)– 20 –ere, ∇ µ is the co-covariant derivative as introduced in section 4.2 including the Weyl gauge field, theaffine connection for coordinate indices (see below for some restrictions) and Ω µ depends on the Lorentzrepresentation of the field the derivative acts on. We also use the abbreviation D A = V µA ( x ) D µ . The spinconnection Ω µ ( x ) must transform like a non-abelian gauge field for local Lorentz transformations, Ω µ ( x ) → Ω ′ µ ( x ) = L R (Λ( x )) Ω µ ( x ) L − R (Λ( x )) − [ ∂ µ L R (Λ( x ))] L − R (Λ( x )) . (4.67)We also write this for an infinitesimal Lorentz transformation Λ AB ( x ) = δ AB + dω AB ( x ) as Ω µ ( x ) → Ω ′ µ ( x ) = Ω µ ( x ) + i dω AB ( x ) (cid:2) M AB R , Ω µ ( x ) (cid:3) − i M AB R ∂ µ dω AB ( x ) . (4.68)This is the transformation rule for a non-abelian gauge field associated to SO(1 , d − Ω µ ( x ) = Ω µAB ( x ) i M AB R , (4.69)where Ω µAB ( x ) is anti-symmetric in the Lorentz indices A and B and now independent of the representation R . Sometimes it is also called spin connection. As an examples we note here the covariant derivative of aLorentz vector with upper index and scaling dimension ∆ A D µ A B ( x ) = ∂ µ A B ( x ) + Ω Bµ C ( x ) A C ( x ) − ∆ A B µ ( x ) A B ( x ) . (4.70)At present, the spin connection Ω Aµ B ( x ) could be of quite general form, as long as it is anti-symmetricin the last two indices. However, in practise, it is most useful to define the spin connection Ω Aµ B such thatthe fully covariant derivative of the tetrad vanishes, D µ V Aν = ∂ µ V Aν + Ω Aµ B V Bν − Γ ρµν V Aρ + B µ V Aν = 0 . (4.71)This leads to a consistent formalism where derivatives of coordinate and Lorentz tensors are compatible.One may solve this relation for the spin connection, leading toΩ Aµ B = − (cid:0) ∇ µ V Aν (cid:1) V νB = − (cid:0) ∂ µ V Aν (cid:1) V νB + (Γ ρµν − B µ δ ρν ) V Aρ V νB . (4.72)From ∂ µ η AB = ∇ µ η AB = ∇ µ ( V µA ( x ) V νB ( x ) g µν ( x )) = 0 one can show that Ω µAB in eq. (4.72) is indeedanti-symmetric as long as the connection Γ ρµν is weakly metric-compatible, in the sense ∇ µ g ρσ = − B µρσ = 0 . (4.73)Proper non-metricity is therefore not permitted in the present formalism. However, both the contorsiontensor C ρµ σ ( x ) and the Weyl gauge field B µ ( x ) are allowed to be non-zero.Finally we note a useful identity for the variation of the spin connection that can be easily derivedfrom (4.72), δ Ω Aµ B ( x ) = − (cid:0) D µ δV Aν (cid:1) V νB + δ (Γ ρµν − B µ δ ρν ) V Aρ V νB . (4.74)We use here the fully covariant derivative D µ and the variation of the affine connection as specified in(4.30) (for vanishing proper non-metricity). Note that in contrast to the spin connection Ω Aµ B ( x ) itself,which is a gauge field, its variation δ Ω Aµ B ( x ) transforms simply as a tensor with one upper and one lowerindex under local Lorentz transformations. Under coordinate transformations both Ω Aµ B ( x ) and δ Ω Aµ B ( x )transform as one-forms. – 21 – .6 Conservation laws in the tetrad formalism Let us now investigate what kinds of conservation-type relations we can obtain from the effective actionΓ[ φ, V, Ω , B ]. We take the latter to depend on matter fields φ ( x ) which can be taken to be local Lorentzvectors, tensors and spinors. In addition the action depends on the tetrad field V Aµ ( x ) which also replacesthe metric everywhere. All derivatives of fields are assumed to be fully covariant derivatives D A whichdepend on the spin connection Ω ABµ and the Weyl gauge field B µ . When we take the spin connectionto be independent of the tetrad and the Weyl gauge field, this is due to the possibility of non-vanishingcontorsion.Because of relation (4.72) or (4.74) the spin connection and can be varied independent of the tetradand the Weyl gauge field only through a variation of contorsion. Even at vanishing physical torsion andcontorsion, it is useful to consider a variation with respect to it. This is similar to varying the metric asdone in section (4.1) even though the latter is subsequently fixed, for example to describe Minkowski space.For stationary matter fields δ Γ /δφ = 0, the variation of the effective action is δ Γ = Z d d x √ g (cid:26) T µA ( x ) δV Aµ ( x ) − S µAB ( x ) δ Ω ABµ ( x ) − d W µ ( x ) δB µ ( x ) (cid:27) . (4.75)The field T µA ( x ) is defined through a variation with respect to the tetrad at fixed spin connection andWeyl gauge field. We will argue below that it is actually the canonical energy-momentum tensor. Thevariation with respect to the spin connection with fixed tetrad defines the spin current S µAB ( x ). Finally thevariation with respect to the Weyl gauge field at fixed tetrad and spin connection defines a field W µ ( x ). Allthree fields T µA ( x ), S µAB ( x ) and W µ ( x ) transform under coordinate transformations and local Lorentztransformations as indicated by their indices. The reason is that the variation δV Aµ ( x ), δ Ω ABµ ( x ) and δB µ ( x ) are all transforming as tensors in this sense and the variation of the action itself must be a scalar.We should also state here that a full variation of the effective action with respect to the tetrad (with thespin connection taken to obey relation (4.72) at fixed contorsion) leads to the energy momentum tensor asa mixed coordinate and Lorentz tensor, while a related variation of the Weyl gauge field gives the dilatationcurrent as in (4.42), δ Γ = Z d d x √ g (cid:26) T µA ( x ) δV Aµ ( x ) − d W µ ( x ) δB µ ( x ) (cid:27) . (4.76)Using (4.74) we can relate the quantities in (4.75) and (4.76) and find T µν ( x ) = T µν ( x ) + 12 ∇ ρ [ S ρµν ( x ) + S µνρ ( x ) + S νµρ ( x )] . (4.77)One can recognize this as the Belinfante-Rosenfeld form of the energy-momentum tensor with the first term T µν ( x ) being the canonical energy-momentum tensor and T µν ( x ) its symmetric relative. Note that theexpression in square brackets in (4.77) is anti-symmetric in ρ and µ . This implies ∇ µ T µν ( x ) = ∇ µ T µν ( x ),so that both tensors are conserved. (This conservation law can be directly obtained from general coordinatetransformations as usual.) In other words, canonical energy-momentum tensor follows from a variationof the action with respect to the tetrad at fixed spin connection, while the symmetric energy-momentumtensor follows from a related variation but at contorsion kept fixed. One also finds for the two versions ofthe dilatation currents defined by (4.75) and (4.76) the relation W µ ( x ) = W µ ( x ) + 2 d S ρµρ ( x ) . (4.78)– 22 –y construction the action is invariant under local Lorentz transformations. We consider now such atransformation in infinitesimal form. The matter fields are still assumed to be stationary, δ Γ /δφ = 0, sothat it suffices to consider the variations of the tetrad and spin connection.We first consider a variation where only the tetrad is being varied, and the spin connection is takenas dependent according to eq. (4.72) at vanishing contorsion. One finds δ Γ = Z d d x √ g T µA ( x ) V Bµ ( x ) δω AB ( x ) . (4.79)Because this must vanish for arbitrary δω AB ( x ) one finds that the energy-momentum tensor is symmetric, T AB ( x ) = T BA ( x ) . (4.80)However, one may also do the calculation in an alternative way where the spin connection is first variedindependent of the tetrad and we use then (4.61) and (4.68), δ Γ = Z d d x √ g (cid:26) T µA ( x ) δV Aµ ( x ) − S µAB ( x ) δ Ω ABµ ( x ) (cid:27) = Z d d x √ g (cid:26) T µA ( x ) δω AB ( x ) V Bµ ( x ) − S µAB ( x ) h δω AC ( x )Ω Cµ B ( x ) − Ω Aµ C ( x ) δω CB ( x ) − ∂ µ δω AB ( x ) i(cid:27) . (4.81)Using partial integration one can rewrite this as δ Γ = Z d d x √ g (cid:20) T BA ( x ) − D µ S µAB ( x ) (cid:21) δω AB ( x ) . (4.82)For this to vanish for arbitrary δω AB ( x ) the expression in square brackets must be symmetric. Because S µAB = − S µBA is anti-symmetric, we find for the divergence of the spin current ∇ µ S µρσ ( x ) = T σρ ( x ) − T ρσ ( x ) . (4.83)This is the conservation-type relation we were looking for. We argue that the spin current S µAB ( x ) shouldbe seen as a non-conserved Noether current associated to an extended symmetry . The transformation in eq.(4.81) is not a full symmetry in the sense of section 2 because a global transformation with D µ δω AB ( x ) = 0does not make the action stationary as long as T AB ( x ) = T BA ( x ). Nevertheless, eq. (4.83) is still a veryuseful identity as long as the right hand side is known. This is indeed the case, because it follows from avariation of the quantum effective action according to eq. (4.75).We emphasize again that the spin current is in general not conserved. What needs to be conserved asa consequence of full Lorentz symmetry in Minkowski space (also including a coordinate transformation)is the sum of spin current and orbital angular momentum current, M µAB ( x ) = x A ( x ) T µB ( x ) − x B ( x ) T µA ( x ) + S µAB ( x ) . (4.84)We assume here D µ x A ( x ) = V Aµ ( x ) (which essentially defines what is meant by x A ( x ) in non-cartesiancoordinates) one has indeed D µ M µAB ( x ) = 0 as a consequence of (4.83) and the conservation law D µ T µA ( x ) = 0. – 23 – .7 General linear frame change transformations Mathematically, the frame bundle allows for changes of basis transformation that are more general thanthe restriction to orthonormal frames we discussed above. The full group of local transformations is thegeneral linear group GL( d ), which contains SO(1 , d −
1) as a subgroup, but encompasses also dilatationsand shear transformations.In the following we discuss first briefly the Lie algebra of GL( d ) and decompose it into the differentgenerators. Subsequently we discuss general frame change transformations and the conservation-typerelations that follow from them.We will consider the general linear group as an extension of the Lorentz group. Accordingly weintroduce in addition to the generators M AB for infinitesimal Lorentz transformations also generators S AB for shear transformations and D for dilatations. Note that we are using indices A and B as for anorthonormal frame to label the generators.The generators for shear transformations are symmetric, S AB = S BA and trace-less, S AB η AB = 0.For d space-time dimensions one has d ( d − / M AB , d ( d + 1) / − S AB and 1generator D . Indeed, these make up the d generators of the general linear group GL( d ). Without thegenerator for dilatations D , the generators M AB together with S AB generate the Lie algebra of the speciallinear group SL( d ).In terms of matrices in the fundamental representation one may take ( M AB ) CD = − i ( η AC δ BD − η BC δ AD ) as usual for the generators of Lorentz transformations, D AB = − iδ AB for the generator of dilata-tions, and ( S AB ) CD = − i ( η AC δ BD + η BC δ AD − (2 /d ) η AB δ CD ) for the generators of shear transformations.The Lie brackets are (cid:2) M AB , M CD (cid:3) = − i (cid:0) η BC M AD − η AC M BD + η BD M CA − η AD M CB (cid:1) , (cid:2) M AB , S CD (cid:3) = − i (cid:0) η BC S AD − η AC S BD + η BD S CA − η AD S CB (cid:1) , (cid:2) S AB , S CD (cid:3) = − i (cid:0) η BC M AD + η AC M BD − η BD M CA − η AD M CB (cid:1) , (cid:2) M AB , D (cid:3) = (cid:2) S AB , D (cid:3) = [ D, D ] = 0 . (4.85)Obviously, M AB and D each generate sub-groups, while the S AB alone do not. The center of the Liealgebra (4.85) is generated by D . It will sometimes be convenient to split GL( d ) into to abelian subgroupof dilatations and the remaining group SL( d ).Previously we have already discussed a group of transformations consisting of SO(1 , d −
1) and thedilatations in terms of orthonormal fields. We will now extend first the indefinite orthogonal group SO(1 , d −
1) to the larger group SL( d ) and subsequently also add the dilatation part.Similar to the discussion of orthonormal frames in section 4.5 we introduce now a frame field orsoldering form that parametrizes the change from a coordinate basis to a more general frame that we maycall an unimodular frame. It can be introduced as a vector valued one form e aµ ( x ) dx µ . The smaller caselatin index a is now belonging to a frame that is in general neither holonomic (induced by a coordinatesystem), nor orthonormal. We may also introduce the inverse frame field such that e aµ ( x ) e νa ( x ) = δ νµ , e aµ ( x ) e µb ( x ) = δ ab . (4.86)The frame field e aµ ( x ) behaves with respect to coordinate transformations very similar as the tetrad V Aµ ( x )and we do not discuss this further. – 24 –he metric g µν ( x ) in the coordinate frame is expressed through the frame field as g µν ( x ) = ˆ g ab ( x ) e aµ ( x ) e bν ( x ) , ˆ g ab ( x ) = g µν ( x ) e µa ( x ) e νb ( x ) . (4.87)Here we introduce the metric in the unimodular frame ˆ g ab ( x ). It has the propertyˆ g = − det g ab ( x ) = 1 , (4.88)but can otherwise we a quite general symmetric matrix. In this sense eq. (4.87) generalizes eq. (4.54).In order to discuss how general linear transformations act on the frame field, let us first excludedilatations, which need a separate discussion because their generator is in the center of the algebra (4.85).Excluding them means here to restrict to transformations with unit determinant, i.e. to restrict fromGL( d ) to SL( d ). Such a special linear transformation acts on the frame field according to e aµ ( x ) → e ′ aµ ( x ) = M ab ( x ) e bµ ( x ) , (4.89)where M ab ( x ) is at every point x a matrix with unit determinant, M ( x ) ∈ SL( d ). Similarly one cantransform other vector fields and tensors with upper indices. Covectors and tensor fields with lower indicestransform with the transpose of the inverse of M ( x ). For example the unimodular metric transforms asˆ g ab ( x ) → ˆ g ′ ab ( x ) = ( M − ) c a ( x )( M − ) db ( x )ˆ g cd ( x ) . (4.90)This makes sure that contractions of upper and lower indices can be done consistently.Two remarks are in order here:(i) Because ˆ g ab ( x ) and its inverse ˆ g ab ( x ) are not invariant symbols with respect to SL( d ), some care isneeded when using them to pull indices down or up.(ii) In a theory with spinor fields one would now have to work with three different frames and correspond-ing indices. Besides the coordinate frame and the general frame one also needs there an orthonormalframe where the Clifford algebra is rooted. An extension of spinor representations from SO(1 , d − d ) is not easily possible. (In principle it is possible to define the operation of general lineartransformations on the Clifford algebra by employing a basis for the latter in terms of p -forms [70],but that has substantial implications we do not discuss further here.) The transition from the or-thogonal frame to the general frame is then mediated by e aB ( x ) = e aµ ( x ) V µB ( x ). We will largelyavoid this technical complication here and assume similar as in section 4.1 that all fermionic fieldshave been integrated out, already. We are then left with fields of integer spin that can be organizedinto scalar, vector and tensor representations under Lorentz transformations. These representationscan be extended to special linear transformations in a rather direct way. Weyl gauge transformations of frame field.
Under a Weyl gauge transformation the frame fieldmust transform analogously to the tetrad (see eq. (4.58)) e aµ ( x ) → e ζ ( x ) e aµ ( x ) . (4.91)Combining this with the SL( d ) transformation in (4.89) leads to the general linear group GL( d ).– 25 – epresentations. We consider now fields φ in some representation R of these generators so that aninfinitesimal transformation reads φ ( x ) → φ ′ ( x ) = φ ( x ) + i dω AB ( x ) M AB R φ ( x ) + i dζ AB ( x ) S AB R φ ( x ) + idζ ( x ) D R φ ( x ) . (4.92)As an example, a vector field ϕ a ( x ) is in the fundamental representation with respect to Lorentz and sheartransformations and would transform for dζ = 0 according to ψ a ( x ) → ψ ′ a ( x ) = ψ a ( x ) + dω ab ( x ) ψ b ( x ) + dζ ab ( x ) ψ b ( x ) . (4.93)We use here dω ab ( x ) = dω AB ( x ) e aA ( x ) e Bb ( x ) etc. Note that Lorentz boosts parametrized by dω ab ( x ) andshear transformations parametrized by dζ ab ( x ) are represented in a closely related way.More formally, the fundamental representation has the generators( M AB F ( x )) c d = − i (cid:2) e Ac ( x ) e Bd ( x ) − e Bc ( x ) e Ad ( x ) (cid:3) , ( S AB F ( x )) c d = − i (cid:2) e Ac ( x ) e Bd ( x ) + e Bc ( x ) e Ad ( x ) − (2 /d ) η AB δ cd (cid:3) . (4.94)Note that the generators depend here on the space-time position x . In a similar way one can find othertensor representations. For example, a covector field would transform as χ b ( x ) → χ ′ b ( x ) = χ b ( x ) − dω ab ( x ) χ a ( x ) − dζ ab ( x ) χ a ( x ) . (4.95)A general ( n, m )-tensor representation of SL( d ) changes under a finite group transformation as φ a ··· a n b ··· b m ( x ) → M a c ( x ) · · · M a n c n ( x )( M − ) d b ( x ) · · · ( M − ) d m b m ( x ) φ c ··· c n d ··· d m ( x ) . (4.96) Dilatations.
Let us now discuss dilatations. Because they are in the center of the algebra (4.85) onecan assign in principle an arbitrary charge to some field ϕ ( x ). Usually this is done such that a scalar fieldwith the (momentum) scaling dimension ∆ ϕ would transform under an infinitesimal dilatation like ϕ ( x ) → ϕ ( x ) − dζ ( x )∆ ϕ ϕ ( x ) . (4.97)In a similar way, any ( n, m )-tensor field in an orthonormal frame would transform under dilatations ac-cording to its scaling dimension ∆ φ (written here for a finite transformation), φ A ··· A n B ··· B m ( x ) → exp( − ζ ( x )∆ φ ) φ A ··· A n B ··· B m ( x ) . (4.98)Dilatations in the unimodular frame are as in an orthonormal frame, because the transition matrix e Ba ( x ) = e µa ( x ) V Bµ ( x ) has the scaling dimension ∆ e Ba = 0. This implies in particular that the metric inthe unimodular frame ˆ g ab ( x ) also has vanishing scaling dimension and for a general tensor one has φ a ··· a n b ··· b m ( x ) → exp( − ζ ( x )∆ φ ) φ a ··· a n b ··· b m ( x ) . (4.99) Covariant derivative.
In order to make derivatives transform in the appropriate representation ofGL( d ), we need to define an appropriately generalized covariant derivative. We will write the latter as D µ = ∂ µ + Ω µ ( x ) . (4.100)– 26 –quation (4.69) is now generalized to Ω µ ( x ) = Ω µAB ( x ) (cid:18) i M AB R ( x ) + i S AB R ( x ) + id η AB D R (cid:19) . (4.101)The general linear connection Ω µAB ( x ) is now not anti-symmetric as the spin connection in the last twoindices any more, but has also a symmetric and trace-less contribution which determines the shear trans-formation sector, as well as a trace which governs dilatations. The trace is directly related to the Weylgauge field by B µ ( x ) = 1 d Ω Aµ A ( x ) . (4.102)For tensor representations of GL( d ) one can directly work withΩ aµ b ( x ) = Ω µAB ( x ) e Aa ( x ) e Bb ( x ) , (4.103)so that for example D µ χ ab ( x ) = ∂ µ χ ab ( x ) + Ω aµ c ( x ) χ c b ( x ) − Ω cµ b ( x ) χ ac ( x ) − B µ ( x )∆ χ χ ab ( x ). Gauge transformations.
The general linear connection is now a gauge field for the group GL( d ). Assuch, it transforms asΩ aµ b ( x ) → Ω ′ aµ b ( x ) = M ac ( x )Ω cµ d ( x )( M − ) db ( x ) − [ ∂ µ M ac ( x )] ( M − ) c b ( x ) . (4.104)For an infinitesimal transformation this readsΩ aµ b ( x ) → Ω ′ aµ b ( x ) = Ω aµ b ( x )+ dω ac ( x )Ω cµ b ( x ) − Ω aµ c ( x ) dω cb ( x ) − ∂ µ dω ab ( x )+ dζ ac ( x )Ω cµ b ( x ) − Ω aµ c ( x ) dζ cb ( x ) − ∂ µ dζ ab ( x ) − ∂ µ dζ ( x ) δ ab = Ω aµ b ( x ) − D µ ( dω ab + dζ ab + dζδ ab ) . (4.105)Similar to the spin connection, the general linear connection Ω aµ b ( x ) should be defined such that thefully covariant derivative of the frame field vanishes, D µ e aν = ∂ µ e aν + Ω aµ b e bν − Γ ρµν e aρ = 0 . (4.106)Note that the right hand side also naturally contains the Weyl gauge field with the right prefactor.This has the advantage that derivatives of tensors can be consistently evaluated in terms of the co-ordinate or the general linear frame. One may solve eq. (4.106) for the general linear connection, leadingto Ω aµ b = − ( ∂ µ e aν ) e νb + Γ ρµν e aρ e νb . (4.107)One may check that (4.107) transforms also correctly, i. e. according to (4.105) and that in contrast to thespin connection (4.72), Ω µab ( x ) is now not anti-symmetric in a and b any more. Also, with eq. (4.28) onecan see that eq. (4.102) in indeed fulfilled.An equation analogous to (4.74) also holds for the variation δ Ω aµ b ( x ), δ Ω aµ b ( x ) = − ( D µ δe aν ) e νb + δ Γ ρµν e aρ e νb . (4.108)– 27 –ne may use here (4.30) for the variation δ Γ ρµν . It is important to note here that the general linearconnection (4.104) is consistently defined also with non-vanishing contorsion C ρµ σ ( x ) and non-metricity B ρµ σ ( x ). In this sense the variation δ Ω aµ b ( x ) in (4.108) is free of algebraic constraints, even if the framefield e aν ( x ) is kept fixed.From eqs. (4.87), (4.106) and (4.17) one finds D µ ˆ g ab ( x ) = − B µab ( x ) . (4.109)The metric ˆ g ab ( x ) is only covariantly constant for vanishing non-metricity. A non-vanishing Weyl gaugefield B µab = B µ g ab is already a deviation from this. Cartan’s structure equations.
For completeness we also note Cartan’s first structure equation fortorsion, T aµν = ∂ µ e aν − ∂ ν e aµ + Ω aµ b e bν − Ω aν b e bµ . (4.110)Using eq. (4.106) one can see that this is indeed in agreement with eq. (4.20). In particular the righthand side vanishes in situations without space-time torsion. Similarly, Cartan’s second structure equationsyields the curvature tensor, R abµν = ∂ µ Ω aν b − ∂ ν Ω aµ b + Ω aµ c Ω cν b − Ω aν c Ω cµ b . (4.111)The Ricci scalar is given by R = g σν e µa e bσ R abµν . Some possible choices for the frame field.
The unimodular frame field e aµ ( x ) can be left open, butit can also be fixed in different ways. Two possibilities are particularly interesting.1. An orthonormal frame is a special case of an unimodular frame. This is obtained by fixing ˆ g ab ( x ) = η ab to the Minkowski metric. The frame field is then a tetrad field, e aµ ( x ) = V aµ ( x ) with all the propertiesdiscussed in sections 4.5 and 4.6.2. One can also set e aµ ( x ) = δ aµ /χ ( x ). Here we have introduced a kind of external dilaton field χ ( x )with scaling dimension ∆ χ = 1 such that it transforms under Weyl transformations according to χ ( x ) → e − ζ ( x ) χ ( x ). Accordingly the frame field has the correct scaling dimension. For this choicethe coordinate metric is of the form g µν ( x ) = ˆ g µν ( x ) /χ ( x ) . Because of ˆ g = − det ˆ g ab ( x ) = 1 one has g ( x ) = − det g µν ( x ) = χ ( x ) − d . Response to local general frame transformations.
Let us now discuss the response of a quantumfield theory to general linear changes of frame. We will again employ the quantum effective action whichdepends on matter field expectation values φ ( x ), the frame field e aµ ( x ) and the general linear connectionΩ aµ b ( x ). In addition it also depends on the metric in the unimodular frame g ab ( x ). Because the metrichas fixed determinant, ˆ g = − det ˆ g ab ( x ) = 1, its variation is trace-free, ˆ g ab ( x ) δ ˆ g ab ( x ) = 0.Because of (4.108) and (4.30) the general linear connection and the frame field e aµ ( x ) are only inde-pendent when contorsion and non-metricity are allowed to vary. In this sense one should understand avariation where Ω aµ b and e aµ are taken to be independent at an intermediate step. We also need to carrythe metric g ab ( x ) because it is needed to construct the coordinate metric g µν ( x ) according to eq. (4.87).Also, g ab and its inverse may of course appear in the effective action.– 28 –e write that effective action as Γ[ φ, e, Ω , ˆ g ] , (4.112)and for stationary matter fields it has the variation δ Γ = Z d d x √ g (cid:26) T µa ( x ) δe aµ ( x ) − S µ ba ( x ) δ Ω aµ b ( x ) + 12 ˆ U ab ( x ) δ ˆ g ab ( x ) (cid:27) . (4.113)This defines a field T µa which must be the canonical energy-momentum tensor in the unimodular frame.This becomes clear when one compares with (4.75) and realizes that ˆ g ab could be kept fixed at the Minkowskiform η ab such that the frame field e aµ is just the tetrad. Moreover, keeping Ω aµ b in (4.113) fixed impliesthen through (4.102) to keep also the Weyl gauge field B µ fixed.Similarly, the field S µ ba must be the the hypermomentum current [43, 44, 46, 47] introduced alreadyin eq. (4.38), now in the unimodular frame. This is because keeping the frame field e aµ and ˆ g ab fixed meansto keep also the coordinate metric g µν fixed and according to (4.108) one has then δ Ω aµ b = δ Γ ρµν e aρ e νb .Finally, ˆ U ab must be the trace-less part of the field U µν ( x ) introduced in (4.38), now in the unimodularframe, ˆ U ab = (cid:2) U µν − (1 /d ) g µν U ρρ (cid:3) e aµ e bν . (4.114)This is because keeping the frame field e aµ and the connection Ω aµ b fixed but varying the metric ˆ g ab is likekeeping the connection Γ ρµ σ fixed and varying only the metric such that g µν δg µν = 0.If instead the variation is done such that (4.108) is obeyed at fixed (vanishing) contorsion and non-metricity we write δ Γ = Z d d x √ g (cid:26) T µa ( x ) δe aµ ( x ) + 12 ˆ T ab ( x ) δ ˆ g ab ( x ) (cid:27) . (4.115)Here T µa must be the symmetric energy-momentum tensor as follows from comparison with (4.76) for δ ˆ g ab = 0. Moreover, when keeping the frame field fixed, δe aµ = 0, we can compare to (4.7) and find thatˆ T ab = (cid:2) T µν − (1 /d ) g µν T ρρ (cid:3) e aµ e bν , (4.116)must be the trace-free part of the symmetric energy-momentum tensor in the unimodular frame.By using (4.108) and eq. (4.5) together with the variation in (4.113) and comparing to (4.115) we findthe following relation T µν = T µν + 14 ∇ ρ [ S ρµν + S µνρ − S µρν − S ρνµ + S νµρ − S νρµ ] , = T µν + 12 ∇ ρ [ S ρµν + S µνρ + S νµρ ] . (4.117)In the second line we have used eq. (4.40) and S ρµν is the spin current. We recognize the Belinfante-Rosenfeld relation between symmetric and canonical energy-momentum tensor as seen before in eq. (4.77).The anti-symmetric part of (4.117) gives eq. (4.83).Similarly we find using (4.41) ˆ T µν ( x ) = ˆ U µν ( x ) + 12 ∇ ρ Q ρµν . (4.118)This is actually eq. (4.47) obtained previously. – 29 – ocal SL ( d ) transformations. In a next step let us consider local SL( d ) transformations. When boththe frame field e aµ and the unimodular metric ˆ g ab are being transformed the coordinate frame metric g µν in (4.87) is invariant. Accordingly also the effective action must be invariant at stationary matter fields.This is indeed the case when T µν as defined by (4.115) is symmetric and when ˆ T µν is its trace-less part. Consequences of Weyl gauge transformations.
Under Weyl gauge transformations the frame fieldstransforms as in (4.91) and the general linear connection as in (4.105). The change in the effective action(4.113) must equal (4.49). We find thus T µµ − √ g ∂ µ (cid:0) √ g S µ aa (cid:1) = U µµ . (4.119)Using (4.39) and (4.77) this can be brought to the form (4.46) as it should be.In summary, in the unimodular frame one can see nicely the full transformation group of the framebundle GL( d ). The associated (non-conserved) Noether currents are the spin current S µρσ , the Weylcurrent W µ and the shear current Q µρσ . The corresponding divergence-type relations are given by eq.(4.83), eq. (4.46) and eq. (4.47). In this section we will discuss an example for an effective action and the resulting construction of thedifferent tensor fields defined in section 4.2. The example is illustrative and rather simple. We take theeffective action for a single real scalar field to beΓ = Z d d x √ g (cid:26) − g µν ∂ µ ϕ∂ ν ϕ − U ( ϕ ) − ξRϕ (cid:27) . (5.1)Here, U ( ϕ ) is the effective potential, R = g σν R σν = g σν R ρσρν is the Ricci scalar and ξ denotes its non-minimal coupling to the scalar field ϕ . For clarity let us note that the Einstein-Hilbert action would be inour conventions S EH = R d d x √ gR/ (16 πG N ). From dimensional analysis the scaling dimension of ϕ followsas ∆ ϕ = ( d − / ξ = ( d − / (4 d −
4) when the action (5.1) has a conformalsymmetry. This value can also be seen as a renormalization group fixed point. On the other side, ξ = 0 isnot a renormalization group fixed point and thus ξ is generated by quantum fluctuations even if it shouldbe absent in the microscopic action.Let us note that as an effective action (5.1) should be seen as an approximation. In particular onecan expect that quantum fluctuation induce more complex kinetic terms, higher order derivatives, moreinvolved couplings to the curvature tensor, as well as non-local terms. Nevertheless, we can use the modelin (5.1) for some illustrations.Let us first consider (5.1) in the context of strictly Riemannian geometry and determine the energy-momentum tensor according to eq. (4.7). This leads to the so-called improved energy-momentum tensor[65], T µν = ∂ µ ϕ∂ ν ϕ − g µν (cid:18) g ρσ ∂ ρ ϕ∂ σ ϕ + U ( ϕ ) + 12 ξRϕ (cid:19) + ξ (cid:0) R µν ϕ + g µν ∇ ρ ∇ ρ ϕ − ∇ µ ∇ ν ϕ (cid:1) . (5.2)– 30 –et us now extend eq. (5.1) to a geometry with general affine connection as discussed in section 4.2.This amounts to taking the connection as independent of the metric or alternatively to introduce contorsion,the Weyl gauge field and proper non-metricity. The action (5.1) becomesΓ = Z d d x √ g (cid:26) − g µν ∇ µ ϕ ∇ ν ϕ − U ( ϕ ) − ξRϕ (cid:27) , (5.3)with the co-covariant derivative acting on a scalar field like ∇ µ ϕ = ( ∂ µ − ∆ ϕ B µ ) ϕ = (cid:18) ∂ µ − d − B µ (cid:19) ϕ. (5.4)The Ricci scalar R is given in eq. (4.34) and its variation in (4.36). We recall also the connection betweenthe Weyl gauge field and the connection in (4.28). Similarly one finds from (4.30) δB µ = 1 d (cid:20) δ Γ ρµ ρ − g ρσ ∇ µ δg ρσ (cid:21) . (5.5)The (non-conserved) tensor U µν and the hypermomentum tensor S µ σρ are defined through eq. (4.38).The variation of the action with respect to the metric at fixed connection yields (evaluated at vanishingcontorsion and non-metricity) U µν = ∂ µ ϕ∂ ν ϕ − g µν (cid:18) g ρσ ∂ ρ ϕ∂ σ ϕ + U ( ϕ ) + 12 ξRϕ (cid:19) + ξR µν ϕ + d − d g µν ∇ ρ ( ϕ∂ ρ ϕ ) . (5.6)Similarly the variation with respect to the connection at fixed metric yields the hypermomentum current, S µ σρ = − d − d δ σρ ∂ µ ϕ − ξg µσ ∂ ρ ϕ + ξδ µρ ∂ σ ϕ . (5.7)The spin tensor can be obtained from this trough eq. (4.40), S µρσ = 12 ( S µρσ − S µσρ ) = − ξg µσ ∂ ρ ϕ + ξg µρ ∂ σ ϕ . (5.8)The shear and dilation current follow through eq. (4.41). We find for the dilatation or Weyl current W µ = (cid:18) ξ d − d − d − d (cid:19) ∂ µ ϕ , (5.9)and for the shear current Q µρσ = − ξg µσ ∂ ρ ϕ − ξg µρ ∂ σ ϕ + ξ d g ρσ ∂ µ ϕ . (5.10)We note that the dilatation current vanishes for the conformal choice ξ = ( d − / (4 d − Conclusions
We have developed here a formalism to determine expectation values as well as correlation functions ofNoether currents from the quantum effective action. These contain immediately all corrections due toquantum fluctuations. Technically, the method works with external gauge fields on which the quantumeffective action depends in addition to expectation values of matter fields.The method is very versatile and can be used in particular for the real, conserved Noether currentsassociated to global symmetries of the quantum effective action. However, it can also be used for a classof transformations that have been called “extended symmetries” [40–42], under which the action is notinvariant but changes in a specific way. More precisely, this change must be proportional to a term that isactually known at the macroscopic level of the quantum effective action in order for the transformation to beuseful in practise. Associated to such “extended symmetries” one finds non-conserved “Noether currents”.Their equation of motion has a form similar to a covariant conservation law but with a non-vanishingknown term on the right hand side.After a general discussion of this construction we turned to applications of these ideas in the contextof space-time geometry. First, the symmetry under general coordinate transformations leads as usual tothe covariant conservation of the symmetric energy-momentum tensor. More interesting are further trans-formations corresponding to changes of basis in the frame- and spin bundle. In particular we discuss localinternal Lorentz transformations (including rotations and boosts), local dilatations or Weyl transforma-tions, and local shear transformations. The associated currents are the spin current, the dilatation or Weylcurrent and the shear current. Together they form a rank-three tensor known as hypermomentum current[43, 44, 46, 47]. The latter can also be understood as the (non-conserved) Noether current associated toGL( d ) transformations in the frame bundle.It is only under special circumstances that real conservation laws arise. For example, the Weyl currentis conserved in the presence of a conformal symmetry (but then typically vanishes). Or the spin current isconserved when the canonical energy-momentum tensor is symmetric. (Our formalism yields an expressionfor the canonical energy-momentum tensor in terms of a variation of the effective action.) The shearcurrent is usually not conserved, except when it vanishes.An interesting application of the insights gained here might concern relativistic fluid dynamics. Whilethe usual formulation builds up on the covariant conservation law for the energy-momentum tensor, ad-ditional equation of motion are available in our formalism, and their connection to the quantum effectiveaction is now understood. It is very interesting that for a given quantum effective action the currentsthemselves are known, as well as their correlation functions, at least in principle. On the other side, thestate dependence of the quantum effective action might be carried to rather good approximation by the(fluid dynamic) degrees of freedom of the energy-momentum tensor, by the matter field expectation values,and additionally by the components of the hypermomentum current. This could lead to a rather power-ful formalism for quantum field dynamics out-of-equilibrium. Understanding how the components of thehypermomentum tensor evolve might be of interest for many situations in non-equilibrium quantum fieldtheory, such as in condensed matter physics, heavy ion collisions, or cosmology.We believe that in particular the dilatation current and the shear current are interesting because theirdivergence-type equation of motion could give the evolution equations for the non-equilibrium degrees offreedom related to bulk and shear viscous dissipation. Moreover, the structure of the equations of motionis such that these equations could actually be causal in the relativistic sense as explained in ref. [21].– 32 –n these regards, our equations are similar to the equations of motion for the so-called divergence-typetheories of relativistic fluid dynamics. We plan to investigate these matters in more detail in a forthcomingpublication.Technically we obtain equations of motion for the components of the hypermomentum tensor byvarying the affine connection independent of the metric, tetrad or frame field. Conceptually this amountsto varying the non-Riemannian parts of space-time geometry, specifically contorsion, the Weyl gauge fieldand proper non-metricity. It is important to note here that for us the non-Riemannian geometry is a purelycalculational device. After the variations are done we evaluate all expressions at vanishing contorsion andnon-metricity, i. e. in the (pseudo) Riemannian geometry of general relativity. However, from the point ofview of theories of modified gravity (beyond Einsteins theory of general relativity), our findings may alsobe of interest. Specifically, it has been argued that the spin current, dilatation current and shear currentare natural source terms to appear in such extended theories of gravity [43–57]. It is therefore useful tounderstand well under which circumstances they are non-zero.In the present paper we have focused entirely on (quantum) field theory. However, in light of ourfindings it might also be interesting to revisit actions for particles (or strings or branes), and to investi-gate wether and how contributions to the shear current, Weyl current and spin current would arise fromvariations in geometry there.On the example of a scalar field theory with non-minimal coupling to gravity we have shown thatall components of the hypermomentum tensor can be non-vanishing. However, they are proportional togradients and might therefore vanish in many equilibrium situations. We believe that quantum fluctuationsindeed induce typically a non-vanishing shear and dilatation current in non-equilibrium situations. Thispoint can be investigated further, for example with the functional renormalization group [3–6], and weplan to do so. Acknowledgments
This work is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)under Germany’s Excellence Strategy EXC 2181/1 - 390900948 (the Heidelberg STRUCTURES ExcellenceCluster), SFB 1225 (ISOQUANT) as well as FL 736/3-1, and by U.S. Department of Energy, Office ofScience, Office of Nuclear Physics, grants Nos. DE-FG-02-08ER41450.
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